On the construction of optimal monotone cubic spline interpolations

In this paper we derive necessary optimality conditions for an interpolatin g spline function which minimizes the Holladay approximation of the energy functional and which stays monotone if the given interpolation data are mon otone. To this end optimal control theory for state-restricted optimal cont rol problems is applied. The necessary conditions yield a complete characte rization of the optimal spline. In the case of two or three interpolation k nots, which we call the local case, the optimality conditions are treated a nalytically. They reduce to polynomial equations which can very easily be s olved numerically. These results are used for the construction of a numeric al algorithm for the optimal monotone spline in the general (global) case v ia Newton's method. Here, the local optimal spline serves as a favourable i nitial estimation For the additional grid points of the optimal spline. Som e numerical examples are presented which are constructed by FORTRAN and MAT LAB programs. (C) 1999 Academic Press.